Reprint of: Weak ε-nets have basis of size O(1/εlog(1/ε) ) in any dimension

Authors:
Nabil H. Mustafa;Saurabh Ray
Affiliations:
Lahore University of Management Sciences, Pakistan;École Polytechnique Fédérale de Lausanne, Switzerland
Venue:
Computational Geometry: Theory and Applications
Year:
2010

Citing 3
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Improved bounds on weak &egr;-nets for convex sets

STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Lectures on Discrete Geometry

Lectures on Discrete Geometry
New Constructions of Weak ε-Nets

Discrete & Computational Geometry

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Abstract

Given a set P of n points in R^d and @e0, we consider the problem of constructing weak @e-nets for P. We show the following: pick a random sample Q of size O(1/@elog(1/@e)) from P. Then, with constant probability, a weak @e-net of P can be constructed from only the points of Q. This shows that weak @e-nets in R^d can be computed from a subset of P of size O(1/@elog(1/@e)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/@e. However, our final weak @e-nets still have a large size (with the dimension appearing in the exponent of 1/@e).